Question

How do you find the limit of $\arctan \left( {{e^x}} \right)$ as $x$ approaches $\infty $?

Answer 1

Hint: This problem deals with finding the limit of the given function, as the variable of the function approaches infinity. In calculus if something is infinite, then we simply mean that there is no limit to its values. We say that as $x$ approaches 0, the limit of $f(x)$ is infinite. Now a limit is a number, or a boundary. So when we say that the limit is infinity, we mean that there is no number that we can name.

Complete step-by-step solution:
Here we have to find the limit of $\arctan \left( {{e^x}} \right)$ where the variable $x$ approaches or tends to $\infty $.
The given function is an inverse trigonometric function of an exponential function, which can also be written as, given below:
$ \Rightarrow \arctan \left( {{e^x}} \right) = {\tan ^{ - 1}}\left( {{e^x}} \right)$
If the highest degree in the numerator and denominator equal, you can use the coefficients to determine the limit, if the highest degree in the numerator is larger than the highest degree of the denominator, the limit will be infinity, and if the highest degree in the denominator is larger than the numerator, then the limit will be equal to zero.
So we know that as $x$ tends to $\infty $, then value of ${e^x}$ is $\infty $, as expressed below:
$ \Rightarrow \mathop {\lim }\limits_{x \to \infty } {e^x} = \infty $
Similarly we know that $\tan \left( {\dfrac{\pi }{2}} \right) = \infty $, hence by taking inverse tangent on both sides:
$ \Rightarrow \dfrac{\pi }{2} = {\tan ^{ - 1}}\left( \infty \right)$
So let $x$ is equal to $\infty $, then ${\tan ^{ - 1}}x = \dfrac{\pi }{2}$, now expressing it in terms of limit:
$ \Rightarrow \mathop {\lim }\limits_{x \to \infty } {\tan ^{ - 1}}\left( x \right) = \dfrac{\pi }{2}$
Now as $x \to \infty $, ${e^x} \to \infty $, so this can be written as:
$ \Rightarrow \mathop {\lim }\limits_{x \to \infty } {\tan ^{ - 1}}\left( {{e^x}} \right) = \dfrac{\pi }{2}$
$\therefore \mathop {\lim }\limits_{x \to \infty } \arctan \left( {{e^x}} \right) = \dfrac{\pi }{2}$

The limit of $\arctan \left( {{e^x}} \right)$ as $x$ approaches $\infty $ is $\dfrac{\pi }{2}$.

Note: Please note that the limits of the arctangent exist at $ - \infty $(minus infinity) and $ + \infty $(plus infinity). The arctangent function has a limit in $ - \infty $ which is $ - \dfrac{\pi }{2}$. The arctangent function has a limit in $\infty $ which is $\dfrac{\pi }{2}$.